Abstract

A kind of homogeneous reaction-diffusion singular predator-prey model with no-flux boundary condition is considered. By using the abstract simplified Hopf bifurcation theorem due to Yi et al. 2009, we performed detailed Hopf bifurcation analysis of this particular pattern formation system. These results suggest the existence of oscillatory patterns if the system parameters fall into certain parameter ranges. And all these oscillatory patterns are proved to be unstable.

1. Introduction

In this paper, we consider the following reaction-diffusion singular predator-prey model:
where and are the population densities of the prey and predator at time and position , and are dimensionless positive parameters; . The underlying spatially homogeneous problem was derived in [1] to model prey-predator interactions in fragile (insular) environments. The spatially structured system (1) supplemented with initial data and no-flux boundary conditions was introduced in [2]. For more information on the system (1), we refer the reader to [1, 2] for great details.

Mathematically, the model was considered by Ducrot and Langlais [3], where the authors first provided a suitable notion of global travelling wave weak solution. They mainly studied the existence of travelling waves solutions for predator invasion in such environments. Under suitable conditions on the diffusion coefficients and on species growth rates they were able to prove that the travelling wave solutions were actually positive on a half line and identically zero elsewhere.

The present paper is targeting to consider the Hopf bifurcations of the reaction-diffusion system (1), by using the simplified Hopf bifurcation theorem due to [4], which is widely used to prove the existence of oscillatory patterns of different kind of pattern formation systems, including Gierer-Minhardt model [5], Degn-Harrison model [6], bimolecular model [7], hair growth controlling model [8], and the Sel’kov model [9].

Our results show that, under suitable choice of system parameters, system (1) will undergo spatially homogeneous and nonhomogeneous oscillatory phenomena. And once the oscillatory patterns exist, they are always unstable. These unstable patterns cannot be observed by numerical simulations; thus, numerical simulations corresponding to our analytical analysis are unavailable in the paper, even though, the analytical results we obtained allow for the clearer understanding of the rich dynamics of the system.

The rest of this paper is structured in the following way. In Section 2, we perform Hopf bifurcation analysis of the system. In Section 3, we draw some concluding remarks to end up our discussion. Throughout the paper, we denote by the set of all the positive integers and .

2. Stability and Hopf Bifurcation Analysis

To begin with, for convenience of our discussion, we copy (1) in the following:

System (2) has a unique equilibrium solution , with
which is in the first quadrant if and only if .

In the following, we always assume that holds. We fix the parameter and choose as the bifurcation parameter.

It follows from [4, 10] that the eigenvalues of are given by these of the operator , which is defined by
whose characteristic equation is
where

We have the following theorem on the stability of the unique positive equilibrium solution .

Theorem 1. Suppose that one of the following conditions holds(1) and ,(2) and and that such that
is satisfied. Then, is always locally asymptotically stable in the reaction-diffusion equation (2).

Proof. On one hand, for any , we have
which implies that always holds for any if
holds.On the other hand, if and , or but holds, then we can obtain that, for any and , we always have . Thus, is always locally asymptotically stable in the reaction-diffusion equation (2).

Now we consider the Hopf bifurcations of the system. According to [4], a point is a Hopf bifurcation point if and only if there exists , such that
and , where is the real parts of the unique pair of complex eigenvalues near the imaginary axis.

Thus, any potential Hopf bifurcation point must be in the interval , where we assume that .

For any Hopf bifurcation point , let be the eigenvalues of near then we have

Then, for any , we have
This implies that the transversality condition is always satisfied for any .

Suppose that holds and define

Then, for any and , we define as the roots of
satisfying , and these points satisfy
and . Clearly and for .

Now we derive a condition from the parameters so that for all and . In fact, from (8) in Theorem 1, it follows that always holds.

Based on the discussions above, we have the following Hopf bifurcation results for the reaction-diffusion model (2).

Theorem 2. Suppose that the constants satisfying the condition (8) and are defined as in (14). Then, for any in , there exist points , , satisfying
such that the system (1) undergoes a Hopf bifurcation at or . Moreover,(1)the bifurcating periodic solutions from are spatially homogeneous, which coincides with the periodic solution of the corresponding ODE system; (2)the bifurcating periodic solutions from are spatially nonhomogeneous.

Next we consider the bifurcation direction and stability of the bifurcating periodic solutions.

Theorem 3. For the system (2), the direction of the Hopf bifurcation at is forward, and the bifurcating (spatial homogeneous) periodic solutions are unstable.

Proof. By Theorem 2.1 of [4], in order to determine the stability and bifurcation direction of the bifurcating periodic solution, we need to calculate . When , we put
where .Now translate (2) into the following system by the translation and , and still let and denote and for the convenience of notation. We have
Following [4], we define
By (2.19) of [4], we have
Following [4], we define
Note that is the complex-value inner product defined as
with , .Then we obtain easily that
Hence it is straightforward to calculate that
By (25), we have
where is the identity matrix. The equalities in (26) lead to
Therefore, we can calculate that
From (13), it follows that , and then by Theorem 2.2 of [4], the direction of the Hopf bifurcation is forward and the bifurcating periodic solutions are unstable since .

Remark 4. (1) The direction of Hopf bifurcations at with can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. Thus, here we did not calculate the bifurcation direction of Hopf bifurcations at with .(2) The spatial nonhomogeneous periodic solutions at with found in Theorem 2 are clearly unstable since the steady state is unstable. (3) Since all the periodic solutions are unstable, the simulations of the oscillatory patterns are unavailable here.

3. Concluding Remarks

In this paper, we considered a kind of diffusive Gaucel-Langleis model. By using the simplified Hopf bifurcation theory due to [4], we were capable of investigating the existence of Hopf bifurcations of the system, which indicates the existence of oscillatory patterns of the system. Our main bifurcation and stability analysis results in the paper can be summarized as follows.(1) If is sufficiently large, say, if holds, then Hopf bifurcations will never be possible. In fact, we can show that if one of the following conditions holds(a) and ,(b) and , and such that is satisfied, then, is always locally asymptotically stable in the reaction-diffusion equation (2) (Theorem 1).(2) However, if is not that large, say, , then, Hopf bifurcation is possible in suitable parameter ranges. In fact, by choosing as the bifurcation parameter, we can obtain that if the constants satisfying the condition (14) and are defined as in (14), then, for any in , there exist points , , satisfying
such that the system (1) undergoes a Hopf bifurcation at or , (Theorem 2). Moreover,(a)the bifurcation direction of Hop bifurcations at is forward and the bifurcating periodic solutions are unstable (Theorem 3),(b)the bifurcation direction of Hopf bifurcations at with can also be calculated as what we did in Theorem 3, using the abstract results due to [4]. However, the calculations are very complicated. This is beyond the scope of this paper, (c)the spatial nonhomogeneous periodic solutions at with found in Theorem 2 are clearly unstable since the steady state is unstable (Remark 4).

Acknowledgment

The authors are grateful to the anonymous referee for the comments and suggestions which definitely led to an improved presentation of the paper.